From these illustrations we derive the following RULE. Q. What do you multiply each denominator by for a new denominator? A. By all the other denominators. Q. What do you multiply each numerator by for a new numerator? A. By the same numbers (denominators) that I multiply its denominator by. Note.-As, by multiplying in this manner, the same denominators are continually multiplied into each other, the process may be shortened; for, having found one denominator, it may be written under each new numerator. This, however, the intelligent pupil will soon discover of himself; and, perhaps, it is best he should. More Exercises for the Slate. Compound fractions must be reduced to simple fractions be tore finding the common denominator; also the fractional parts of mixed numbers may first be reduced to a common denominator, and then annexed to the whole numbers. 7. Reduceof and to a common denominator. 8. Reduce 14 and & to a common denominator. A. 11, 29. A. 1418, 2. 9 Reduce 103 and of to a common denominator. A. 1088, 8. 10. Reduce 8 and 144 to a common denominator. A. 8777, 1474 Notwithstanding the preceding rule finds a common denominator, it does not always find the least common denominator But, since the common denominator is the product of all the given denominators into each other, it is plain, that this product ( XLII.) is a common multiple of all these several denominators, consequently, the least common multiple found by ¶ XLII will be the least common denominator. 11. What is the least common denominator of, and = of 64, the new numerator, written over the 6, = . Hence, to find the least common denominator of several fractions, find the least common multiple of the denominators, for the common denominator, which, multiplied by each fraction, will give the new numerator for said fraction. 12. Reduce and to the least common denominator. 14. Reduce 14 and 13 to the least common denominator. A. 1419, 13. Fractions may be reduced to a common, and even to the least common denominator, by a method much shorter than either of the preceding, by multiplying both the terms of a fraction by any number that will make its denominator like the other denominators, for a common denominator; or by dividing both the terms of a fraction by any numbers that will make the denominators alike, for a common denominator. This method oftentimes will be found a very convenient one in practice. 15. Reduce and to a common, and to a least commor. denominator. ×2; then § and 2); then and common denominator, A. least common denominator, A. In this example, both the terms of one fraction are multiplied, and both the terms of the other divided, by the same number consequently, (¶ XXXVII.) the value is not altered. 16. Reduce and to the least common denominator. A. 12, 12. 17. Reduce and to the least common denominator. 00 ¶ XLIV. 1. A father gave money to his sons as follows; to William of a dollar, to Thomas, and to Rufus , how much is the amount of the whole? How much are ,, and, added together? 2. A mother divides a pie into 6 equal pieces, or parts, and gives to her son, and to her daughter; how much did she give away in all? How much are and added together? 3. How much are † +3+§? 6. How much are 2%+2%+2%? When fractions like the above have a common denominator expressing parts of a whole of the same size, or value, it is plain, that their numerators, being like parts of the same whole, may be added as in whole numbers; but sometimes we shall meet with fractions, whose denominators are unlike, as, for example, to add and together. These we cannot add as they stand; but, by reducing their denominators to a common denominator, by ¶ XLIII., they make and, which, added together as befce, make §, Ans. 1. Bought 3 loads of hay, the first weighing 192 cwt., the second 20 cwt., and the third 22 cwt.; wat was the weight of the whole ? ,,, reduced to a common denominator, are equal to 5, 48 and 3 these, joined to their respective whole numbers, give the following expressions, viz. 9 2. How much is of, and, added together?of=; then and, reduced to a common denominator, give 24 and 14, which, added together as before, give 24=124, Ans. From these illustrations we derive the following RULE. Q. How do you prepare fractions to add them? A. Reduce compound fractions to simple ones, then all the fractions to a common or least common denominator. Q. How do you proceed to add? · A. Add their numerators. More Exercises for the Slate. 3. What is the amount of 16 yards, 17 yds. and 3 yards 2. Harry had of a dollar, and Rufus ; what part of a dollar has Rufus more than Harry? How much does from leave? 3. How much does 18 from 13 leave? 4. How much does 18 from 5. How much does 6. How much does leave? From the foregoing examples, it appears that fractions may be subtracted by subtracting their numerators, as well as added, and for the same reason. 1. Bought 20 yards of cloth, and sold 15 yards; how much remained unsoid? In this example, we cannot from, but, by bor take makes 16; then, 16 from 20 leaves 4, which, joined with 12, makes 41, Ans. 2. From taket. and t, reduced to a cominon denomi nator, give and ; then, from leaves 3o, Ans. From these illustrations we derive the following Q. What is the rule? RULE. A. Prepare the fractions as in addition, then the difference of the numerators written over the denominator, will give the difference required. More Exercises for the Slate |